Question

The sequences x₁ [n] and x₂ [n] are each of length N, but can be zero-padded out to length 2N and their length-2N DFTs taken to become X₁(k) and X₂(k). T

he sequence y[n] is formed from them as y[n]={x₁[0], x₁[1], x₁[2],.....,x₁(N - 1), x₂ [0], x₂ [1], x₂ [2],...., x₂ (N-1)
​and (clearly) has length 2N. Select and justify one of the length-2N DFT's Y(k) below.
X₁(k) + Wᵏ₂ₙ X₂ 0 ≤ k < N
(a) Y(k)=
X₁(k) - Wᵏ₂ₙ X₂ N ≤ k < 2N

​(b) Y(k)= X₁(N + K) + X₂(N-K)

(c) Y(k)= X₁(k) - Wᵏ₂ₙ X₂(N-K)

X₁(k) + X₂(K) k even
(d) Y(k)=
X₁(k) - X₂(K) k odd

​e) none of these

Answer 1

The DFT Y(k) of the sequence y[n], formed by concatenating x1[n] and x2[n] both zero-padded to 2N, is given by Y(k) = X1(k) + W^kN X2(k) for the first half and Y(k) = X1(k - N) - W^kN X2(k - N) for the second half, utilizing DFT properties.

Step-by-step explanation:

The correct length-2N DFT of the sequence y[n] that is formed by concatenating x1[n] and x2[n] of length N, both zero-padded to length 2N, can be found by applying discrete Fourier Transform properties. The sequence y[n] is clearly composed of two parts, where the first N samples are from x1[n] and the latter N samples are from x2[n]. The property of the DFT of concatenated sequences tells us that the expression for Y(k) would exhibit periodicity and involve twiddle factors to account for the phase shifts introduced by concatenation.

Option (a) suggests the correct form for Y(k):
Y(k) = X1(k) + W^kN X2(k) for 0 ≤ k < N, and
Y(k) = X1(k - N) - W^kN X2(k - N) for N ≤ k < 2N.
This is representative of the DFT for concatenated sequences where the second sequence is modulated by a phase factor of W^kN to reflect the shift in the time domain.